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DISCOVERY OF PARTHENON MATH
By Prof. L. Kaliambos (Kaliambos-Natural Philosophy) February 9 , 2016 In my papers “Discovery of Golden Section in Parthenon” and "Discovery of Phi in Discobolus” I showed that the mathematics of the golden section was used not only in the construction of the Cheops pyramid (2560 BC) but also in Parthenon and in Hephaestion tomb. Therefore the math of golden section was well known to Myron, Phidias, Mnecicles , and Dinocrates. Neverteless, for understanding better the use of golden section based on golden rectangles in Parthenon built by the ancient Greeks from 447 to 438 BC you can see in Google “The Parthenon and Phi, the Golden Ratio”.This photo is from the interview I gave to the author of Spiritual Thessaly, Dimitra Bardani, through the TV Thessally (Greece) about my discovey of Parthenon math. (ΑΝΑΚΑΛΥΨΗ ΤΩΝ ΜΑΘΗΜΑΤΙΚΩΝ ΤΟΥ ΠΑΡΘΕΝΩΝΑ) Plato (circa 428 BC - 347 BC), in his views on natural science and cosmology presented in his “Timaeus,” considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos. In nature this golden section as a principle may be observed in the arrangements of leaves on a twig, petals on a flower and the arms of the starfish. The ancient Greeks considered a rectangle whose sides are in this ratio to be aesthetically the most pleasing of all rectangles and constructed their buildings on this principle. In fact, in my paper “Discovery of Φ and π in Giza great pyramid ” I showed that both the Pi = π = C/d =3.1416 and the Phi = Φ = (1+50.5)/2 = 1.618034 were used in the construction of Cheops pyramid . Also in my paper “Relation of Pi to Phi and mystic numbers” I showed that such mystic numbers like Φ, 3, π, and 12 were used in the construction of the Mathematical tomb of Hero Hephaestion (320 BC). Golden Section Phi = Φ = (1 + 50.5)/2 is obtained by dividing a line into two parts (α and β) such that the square of the first part is equal to the product of the whole segment (α+β) and the second part. That is α2 = (α +β )β or (α +β )/α = α/β = Φ One method for finding the value of Φ is to write α = Φ and β = 1 (unit length). Therefore (Φ+1)/Φ = Φ/1 or Φ + 1 = Φ2 which can be rearranged to Φ2 - Φ -1 = 0. Today it is well known that in a quadratic formula of the following general form aΦ2 + bΦ + c = 0 the value (positive ) of Φ is given by Φ = [ -b + (b2 -4ac)0.5] / 2a Since a = 1, b = -1, and c = -1 one gets Φ = (1+ (1+4)0.5] / 2 = (1+50.5) / 2 = 1.6180339887…. Such a golden number is obvious in my discovery of the golden section in Caryatids of Erechtheion. The Egyptians used the Phi = Φ = 1.6180339887…) in the design of the Great Pyramids and they thought that the golden ratio was sacred. Therefore, they used the golden ratio when building temples and places for the dead. The Egyptians were aware that they were using the golden ratio Φ. However, one should ask how the ancient Egyptians were able to find the solution of the quadratic formula Φ2 - Φ -1 = 0 . The study of this called algebra goes to the antiquity. Recent discoveries have shown that Babylonians and Egyptians solved problems in algebra , although they had no symbols for variables. They used only words to indicate such numbers, and for that reason their algebra has been referred to as theoretical algebra. The Ahmes Papyrus, an Egyptian scroll going back to 1600 BC has a number of problems in algebra, in which the unknown is referred to as a hau, meaning “a heap”. Also practically the so -called Pythagorean theorem (6thcentury BC) was well known to Babylonians and Egyptians. Thus writing 1 + Φ = Φ2 as (1)2 + (Φ0.5)2 = Φ2 one sees that the 1 (unit length) should be the radius r of a cone pyramid, while Φ0.5 = h (height) and Φ = L (slant height). In this case the circumference C =2π because r = 1. In other words such a cone pyramid was believed to be a sacred pyramid because it includes the mystic numbers Phi = Φ and Pi = π . Also the great pyramid of Cheops was believed to be a sacred square pyramid, because a theoretical cone pyramid was inscribed in the square pyramid. Another ratio does appear throughout most of the Parthenon. This equals a 9 : 4 ratio which can be related to the sacred Pythagorean rectangles of the numbers 3, 4, and 5. According to the history of Greek People (Volume Γ2 page 282) the dimensions of the Parthenon are Height z = 13.724 m . Width y = 30.88 m. Length x = 69.48 m On the exterior, the Doric columns measure 1.9 m in diameter and are 10.4 m high. The corner columns are slightly larger in diameter. The Parthenon had 46 outer columns and 23 inner columns in total, each column containing 20 flutes. (A flute is the concave shaft carved into the column form.) The stylobate has an upward curvature towards its centre of 60 millimetres on the east and west ends, and of 110 millimetres on the sides. The roof was covered with large overlapping marble tiles known as imbrices and tegulae. According to the "Mathematics and architecture-Wikipedia" the columns might be supposed to lean outwards, but they actually lean slightly inwards so that if they carried on, they would meet about a mile above the centre of the building. In general for the construction of the Temple of Athena Parthenos the mathematical principles of proportion of the Pythagoreans were used like the basic rectangle of sides 3 and 4 giving the diagonal of 5 as 32 + 42 = 52 Moreover a rectangle of sides 9 : 4 was constructed from three rectangles of sides 3 and 4 with diagonal 5. In this case the total ratio is given by 9 : 4 = (3/4 + 3/4 + 3/4 ) Moreover the ratio 92 : 42 = 5.0625 Such a similarity also meant that the 3 : 4 : 5 Pythagorean triangle could be used to good effect to ensure that right angles in the building were accurately determined. Now let us look briefly at the dimensions of the Parthenon to see how the lengths conform to the mathematical principles of proportion of the Pythagoreans. In 480 BC the Acropolis in Athens was totally destroyed by the Persians in the Second Persian War. To understand the timescale, let us note that this was about the time of the death of Pythagoras. After the Greek victory over the Persian at Salamis and Plataea the Greeks did not begin the reconstruction of the city of Athens for several years. Only after the Greek states ended their fighting in the Five Years' Truce of 451 BC did the conditions exist to encourage reconstruction. Pericles, the Head of State in Athens, set about rebuilding the temples of the Parthenon in 447 BC. The architects Ictinus and Callicrates were employed, as was the sculptor Phidias. Here the ratios 9 : 4 = 2.25 and 92 : 42 = 5.0625 of three Pythagorean rectangles were fundamental to the construction. In other words a basic rectangle of sides 9 : 4 was constructed from three rectangles of sides 3 and 4 with diagonal 5. Under this condition we write y/z = 30.88/13.724 = 2.25 = 9/4 or y = (9/4)z x/y = 69.48/30.88 = 2.25 = 9/4 or x =(9/4)y x/z = 69.48/13.724 = 5.0625 = 92/42 or x = (92/42)z = (5.0625)z Here we see that x is 5.0625 times greater than z =13.724 m Whereas using the golden number Φ = (1+ 50.5)/2 we could write y/z = Φ , x/y = Φ and x/z = Φ2 = ( 1+50.5)2/22 = 2.618 which means that x should be only 2.618 times greater than z. Therefore a construction based on the golden numbers 1.618 and 2.618 cannot give impressive dimensions. Whereas the use of the Pythagorian rectangles give the Parthenon structure under a perfect harmony. In this case we take the greatest common denominator of these measurements to arrive at the ratios height : width : length = 16 : 36 : 81 Then the length of the Temple is 92 modules, its width is 62 modules and its height is 42 modules. The module length is used throughout, for example the overall height of the Temple is 21 modules, and the columns are 12 modules high. The naos, which in Greek temples is the inner area containing the statue of the god, is 48.285 m long and 21.46 m wide and which again is in the ratio 48.285/21.46 = 2.25 = 9 : 4. Also we see the amazing fact that the distance between their axes is 4.293 m, and columns are 1.908 m in diameter and again the ratio 4.293/1.908 = 2.25 = 9 : 4 is being used. To conclude we see that in Parthenon there are many geometric constructions of the Golden Ratio based on a golden rectangle whose ratio of the longer side to the shorter side is Φ = (1 + 50.5)/2 = 1.618034. Also in Caryatids of Acropolis I discovered that the method of designing the golden section is the same as that of the Caryatids in Hephaestion tomb. Nevertheless for the design of the whole temple having impressive dimensions under a harmony the Athenians decided to construct it by using the mathematics of Pythagorean rectangles with the ratio 9 : 4. Although it looks simple in construction, it has a harmony about it which is based on geometrical mathematics. Whereas the very successful use of the golden section in sculptures cannot give impressive dimensions of the whole temple. Category:Fundamental physics concepts